Optimal. Leaf size=118 \[ \frac{16 \left (a e^2-b d e+c d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0416106, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {722, 636} \[ \frac{16 \left (a e^2-b d e+c d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 722
Rule 636
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{\left (8 \left (c d^2-b d e+a e^2\right )\right ) \int \frac{d+e x}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac{2 (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{16 \left (c d^2-b d e+a e^2\right ) (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.816332, size = 190, normalized size = 1.61 \[ \frac{2 \left (12 b (d-e x) \left (2 a^2 e^2+a c (d-e x)^2+2 c^2 d^2 x^2\right )-8 \left (3 a^2 c e \left (d^2+e^2 x^2\right )+2 a^3 e^3-3 a c^2 d x \left (d^2+e^2 x^2\right )-2 c^3 d^3 x^3\right )-6 b^2 \left (d^2-6 d e x+e^2 x^2\right ) (a e-c d x)+b^3 \left (-9 d^2 e x-d^3+9 d e^2 x^2+e^3 x^3\right )\right )}{3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.047, size = 296, normalized size = 2.5 \begin{align*} -{\frac{24\,abc{e}^{3}{x}^{3}-48\,a{c}^{2}d{e}^{2}{x}^{3}-2\,{b}^{3}{e}^{3}{x}^{3}-12\,{b}^{2}cd{e}^{2}{x}^{3}+48\,b{c}^{2}{d}^{2}e{x}^{3}-32\,{c}^{3}{d}^{3}{x}^{3}+48\,{a}^{2}c{e}^{3}{x}^{2}+12\,a{b}^{2}{e}^{3}{x}^{2}-72\,abcd{e}^{2}{x}^{2}-18\,{b}^{3}d{e}^{2}{x}^{2}+72\,{b}^{2}c{d}^{2}e{x}^{2}-48\,b{c}^{2}{d}^{3}{x}^{2}+48\,{a}^{2}b{e}^{3}x-72\,a{b}^{2}d{e}^{2}x+72\,abc{d}^{2}ex-48\,a{c}^{2}{d}^{3}x+18\,{b}^{3}{d}^{2}ex-12\,{b}^{2}c{d}^{3}x+32\,{a}^{3}{e}^{3}-48\,d{e}^{2}{a}^{2}b+48\,{a}^{2}c{d}^{2}e+12\,a{b}^{2}{d}^{2}e-24\,abc{d}^{3}+2\,{b}^{3}{d}^{3}}{48\,{a}^{2}{c}^{2}-24\,ac{b}^{2}+3\,{b}^{4}} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 8.75005, size = 783, normalized size = 6.64 \begin{align*} \frac{2 \,{\left (24 \, a^{2} b d e^{2} - 16 \, a^{3} e^{3} -{\left (b^{3} - 12 \, a b c\right )} d^{3} - 6 \,{\left (a b^{2} + 4 \, a^{2} c\right )} d^{2} e +{\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \,{\left (b^{2} c + 4 \, a c^{2}\right )} d e^{2} +{\left (b^{3} - 12 \, a b c\right )} e^{3}\right )} x^{3} + 3 \,{\left (8 \, b c^{2} d^{3} - 12 \, b^{2} c d^{2} e + 3 \,{\left (b^{3} + 4 \, a b c\right )} d e^{2} - 2 \,{\left (a b^{2} + 4 \, a^{2} c\right )} e^{3}\right )} x^{2} + 3 \,{\left (12 \, a b^{2} d e^{2} - 8 \, a^{2} b e^{3} + 2 \,{\left (b^{2} c + 4 \, a c^{2}\right )} d^{3} - 3 \,{\left (b^{3} + 4 \, a b c\right )} d^{2} e\right )} x\right )} \sqrt{c x^{2} + b x + a}}{3 \,{\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} +{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} +{\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \,{\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.12298, size = 474, normalized size = 4.02 \begin{align*} \frac{{\left ({\left (\frac{{\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + 24 \, a c^{2} d e^{2} + b^{3} e^{3} - 12 \, a b c e^{3}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{3 \,{\left (8 \, b c^{2} d^{3} - 12 \, b^{2} c d^{2} e + 3 \, b^{3} d e^{2} + 12 \, a b c d e^{2} - 2 \, a b^{2} e^{3} - 8 \, a^{2} c e^{3}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{3 \,{\left (2 \, b^{2} c d^{3} + 8 \, a c^{2} d^{3} - 3 \, b^{3} d^{2} e - 12 \, a b c d^{2} e + 12 \, a b^{2} d e^{2} - 8 \, a^{2} b e^{3}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac{b^{3} d^{3} - 12 \, a b c d^{3} + 6 \, a b^{2} d^{2} e + 24 \, a^{2} c d^{2} e - 24 \, a^{2} b d e^{2} + 16 \, a^{3} e^{3}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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